The Delta Conjecture

James Haglund (1) , Jeffrey B. Remmel (2) , Andrew Timothy Wilson (1)

1 Department of Mathematics [Philadelphia]
2 MATH - UC San Diego - Department of Mathematics [Univ California San Diego]

We conjecture two combinatorial interpretations for the symmetric function ∆eken, where ∆f is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both interpretations can be seen as generalizations of the Shuffle Conjecture, a statement originally conjectured by Haglund, Haiman, Remmel, Loehr, and Ulyanov and recently proved by Carlsson and Mellit. We show how previous work of the second and third authors on Tesler matrices and ordered set partitions can be used to verify several cases of our conjectures. Furthermore, we use a reciprocity identity and LLT polynomials to prove another case. Finally, we show how our conjectures inspire 4-variable generalizations of the Catalan numbers, extending work of Garsia, Haiman, and the first author.

Domaines: Combinatoire [math.CO]

James Haglund, Jeffrey B. Remmel, Andrew Timothy Wilson. The Delta Conjecture. 28-th International Conference on Formal Power Series and Algebraic Combinatorics, Simon Fraser University, Jul 2016, Vancouver, Canada. ⟨10.46298/dmtcs.6384⟩. ⟨hal-02168294⟩